Reading

Wilf's book on generating
functions is a good secondary reference, available for free here. The course synopsis below will contain other specific suggestions as the course progresses.

Synopsis

A week-by-week record of what's going on:

9-8: Introduction and overview. Our focus will be mostly on
counting. What,
though, constitutes a solution to a counting problem?

9-11,
9-13, 9-15: permutations and combinations. We can organize some
classical counting problems in terms of additive and multiplicative
principles. \(k\)-element sequences of entries
from \([n]=\{1,2,...,n\}\) can also be thought of as functions
\(f\colon[k]\to [n]\). From this point of view permutations are
the injective functions. The discussion extends to permutations
of multisets, which we see can be counted by multinomial
coefficients. Subsets of \([n]\) are in one-to-one correspondence
with permutations of a multiset with just two kinds of element.
This gets us back to classical binomial coefficients.

9-18: the pigeonhole principle 9-20: fancy variations and applications like the Erdos-Szekeres Theorem

9-22: generating permutations: the Johnson-Trotter algorithm.

Syllabus

This is an intermediate course in enumerative combinatorics, the
study
of counting. There are not many formal prerequisites, but you
will enjoy the course best if you have some enthusiasm for
problem-solving and hands-on math. We will review the basics -- how to
count
permutations and combinations of labelled and unlabelled objects.
We will see how to use formal power series (also known
as generating functions) to solve counting problems easily and
systematically. The usual topics include:

the pigeonhole principle

permutations and combinations of sets and multisets

introducing formal power series

ordinary and exponential generating functions

solving recurrence relations

counting graphs and trees

Joyal's theory of species

counting in the presence of symmetry (Polya theory) or
Lagrange
inversion

Assignments

Learning the art of counting requires, above all, practice.
Accordingly, there
will be regular homework assignments. This is the most important
part of the course. Please note that no late assignments will be
accepted. See the homework page for an
up-to-date
list.

Some of the assignment problems will be routine, and some will take
some
thought. Collaborating with other people can add a lot to the
experience of doing math, and I encourage you to do so.
(Research-level mathematics can be done alone, but is probably more
often done in groups of two or three.) Just make sure to write
your
own solutions, your own way, and to acknowledge any debts you may
have. Ask me if in doubt, since presenting the work of others as your own constitutes a serious academic offence.

Sometimes it can be useful to use some symbolic computation software,
for example to evaluate a few terms of a power series. Try Maple
or Mathematica, if you have access or familarity. You can also
use Sage, an open-source
symbolic
computation tool, online and for free. For example, create a Sage
notebook, and enter the following:

var('t')
f = e^(e^t-1)
f.taylor(t,0,10)

This will give you the first ten terms of the exponential generating
function for the Bell numbers, which we will learn about in early November.

Exams

There will be one midterm which we will schedule at the start of week 2.

Math 9043a

The MSc version of this course includes slightly
different homework problems, and an additional self-directed
written
project,
to be chosen at the start of term. In this case, the evaluation is
weighted
as 30% final exam; 25% midterm; 25% assignments; 20% project. The
project is due on the first Monday after the last lecture.

Further information

Academic dishonesty:
Scholastic offences are taken seriously and students are directed to read the official policy.

Accessibility Statement:
Please contact the course instructor if you require material in an
alternate format or if you require any other arrangements to make this
course
more accessible to you. You may also wish to contact Services for
Students with Disabilities (SSD) at 661-2111 ext. 82147
for any specific question regarding an accommodation.

Support Services:
Learning-skills counsellors at the Student Development Centre
are ready to help you improve your learning skills.
Students who are in emotional/mental distress should refer to
Mental Health@Western
for a complete list of options about how to obtain help.
Additional student-run support services are offered by the
USC.
The website for Registrarial Services is
http://www.registrar.uwo.ca.

Eligibility:
You are responsible for ensuring that you have successfully completed
all course prerequisites and that you have not taken an antirequisite
course.
Unless you have either the requisites for this course or written special permission
from your Dean to enroll in it, you may be removed from this course and it will be
deleted from your record. This decision may not be appealed. You will receive
no adjustment to your fees in the event that you are dropped from a course for
failing to have the necessary prerequisites.

Postal Address

Department of Mathematics
Middlesex College
University of Western Ontario
London, ON N6A 5B7
Canada